Preheat the oven to 375 degrees F. In a large ovenproof skillet or a roasting pan over two burners, heat 2 tablespoons butter over medium-low heat until melted and browned. Stir in the onion and cook, stirring occasionally, until caramelized, about 15 minutes. Transfer to a plate.
Add the remaining 2 tablespoons butter to the pan and cook until browned but not burned. Add the mushrooms and sage and cook until caramelized, about 10 minutes. Stir in the cooked onions and the chicken broth and bring to a boil over medium heat. Cook until most of the broth has evaporated, about 10 minutes; season with salt and pepper.
Push the mushroom mixture to the side and place the trout fillets skin side down in the pan. Spoon the mushroom mixture over the fish and bake until cooked through, 10 minutes.